Analytic sets and Borel isomorphisms
In this note we give some new characterizations of distributivity of a nearlattice and we study annihilator-preserving congruence relations.
The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice of all deductive systems on and every maximal deductive system is prime. Complements and relative complements of are characterized as the so called annihilators in .
We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra . We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice of all deductive systems on . Moreover, relative annihilators of with respect to are introduced and serve as relative pseudocomplements of w.r.t. in .
A new ⋄-like principle consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that is consistent with CH and that in many models of = ω₁ the principle holds. As implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....
It is consistent that Assuming four strongly compact cardinals, it is consistent that
Shelah’s pcf theory describes a certain structure which must exist if is strong limit and holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially...